Leading indicators of instability based on different elements of the covariance matrix (S_y), including the maximum (in absolute value) element, <i>Max</i>[S_y], the difference between <i>Max</i>[S_y] and <i>Min</i>[S_y], the element of S_y corresponding to the most connected, least connected, or highest eigenvector centrality (24) network node.

<p>Random (l<i>eft</i>) and scale free (right) (30) network generated with <i>N</i> = 50 and <i>C</i> = 0.1 (main panels) and <i>N</i> = 0.1 and <i>C</i> = 0.5 (insets). Instability (i.e., decrease in <i>Max</i>[<i>Re</i>(λ)]) is attained by increasing the interaction strength <i>p</i> (mean field case). The figures represent average behavior over 100 realizations.</p

<i>Max</i> [S_y]as a leading indicator of instability in a “mean field” network with constant interaction intensity (in absolute value), <i>p</i>.

<p>Instability is attained by increasing <i>p</i> (main panel A, with <i>N</i> = 20, <i>C</i> = 0.2) or <i>C</i> (inset B, with <i>N</i> = 20, and <i>C</i> increasing from 0.1 to 1) with different network structures. The figures represent average behavior over 1000 realizations.</p

Distribution of the correlation, ρ_K, between <i>Max</i>[S_y] and the parameter <i>p</i>, after 1000 realizations for the full disordered (not mean-field) case.

<p>If ρ_K is significant (p-value<0.05) and ρ_K>0.5 the increase in <i>Max</i>[S_y] is interpreted as an early warning sign. We calculate these detection statistics for several realizations of each network structure and determine the probability of detecting the early warning sign of instability. We consider eleven different network architectures typical of ecological or social networks, including random (R), predator-prey (PP), cascade (Casc), compartmentalized (Comp), mutualistic (M), bipartite (Bip), nested (N), nested with competition (N+C), scale free (SF), and small world (SW). These networks have different structures for the adjacency matrix and different combination of interaction types, i.e (++) mutualistic, (+−) antagonistic, (−−) competitive or a combination of them (See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0101851#pone.0101851.s019" target="_blank">Materials S1</a> for more details).</p

The 10 highest and lowest specialization food in the year 2011.

<p>The 10 highest and lowest specialization food in the year 2011.</p

Topology of bipartite country-food product networks.

<p>(a) The nestedness (measured by the NODF given by [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0165941#pone.0165941.ref028" target="_blank">28</a>] <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0165941#pone.0165941.e007" target="_blank">Eq (1)</a>) of country-food production bipartite empirical networks (black point) and corresponding 100 randomly assembled networks of the same size and connectivity (blue points denote the mean and the bands are one standard deviation above and below the mean) for each year; (b) The cumulative in-degree and (c) out-degree distribution of the country-food production network for the years 1992-2011 (gray dots). These are distributed consistently with Weibull distribution (red think line) with parameters <i>α</i> = 2.510, <i>β</i> = 30.000, <i>μ</i> = −0.387 and with <i>α</i> = 1.774, <i>β</i> = 32.230, <i>μ</i> = 0.609, respectively.</p

Distribution of country fitness and food specialization.

<p>The cumulative distribution of (a) country fitness and (b) production specialization. The country fitness and food specialization variables reach a fat tail distribution (Log-Normal with parameters <i>μ</i> = −0.607839, <i>σ</i> = 1.18597 and <i>μ</i> = −2.37283, <i>σ</i> = 2.40229, respectively) independently of the algorithm initial condition.</p

Emergent correlation in food production and import.

<p>(a) Example of a typical sub-tree from the MSF of the country-food production network for the year 2011. This sub-tree reflects the correlation in the food production among these countries, highlighted by the corresponding correlation matrix plot; (b) Example of a typical sub-tree from the MSF of country-food import network. The detected correlation in food importion among these specific countries is due to the fact that a large fraction of imports in these countries is comprised of three food commodities: maize, wheat and rice (see corresponding table).</p

Relationship between country GDP and fitness.

<p>(a) The static country fitness—GDP per capita plane of the year 2011. We do not find significant correlation between these two quantities; for example Singapore has high GDP per capita but low fitness. On the other hand Bolivia has high fitness (world leader producer of Quinoa), but low GDP per capita. (b) The dynamical evolution of country fitness and GDP per capita of China, India and Brazil from 1992 to 2011. As time goes on, the GDP per capita of these countries increase, but the fitness of Brazil and India are constant on average while China has increased.</p

Bipartite country-food production network and its projection networks.

<p>A simple example showing how the two projection networks arise from the bipartite country-food production network (a) which describes the different food commodities produced by each country. (b) Food-food projection network obtained from the bipartite graph shown in (a). In this projection, nodes represent different food commodities which are connected to each other if they both are produced by the same country. (c) Country-country network obtained from the bipartite graph shown in (a) where nodes represent countries and countries are connected if they produce the same food commodity.</p

The 10 highest and lowest fitness countries in the year 2011.

<p>The 10 highest and lowest fitness countries in the year 2011.</p